Exercise 2.1
Consider the two states |psi
angle and |chi
angle,
where |phi_{1}
angle and |phi_{3}
angle are orthonormal.
(a) Calculate (langlepsi|psi
angle),(langlechi|chi
angle),(langlepsi|chi
angle),(langlechi|psi
angle), and infer (langlepsi+chi|psi+chi
angle). Are the scalar products (langlepsi|chi
angle) and (langlechi|psi
angle) equal?
Hint: Find the corresponding bras langle and langle first. Then take the scalar (inner) product, remembering that langlephi_{1}|phi_{1}
angle and all other scalar (inner) products are zero.
Also, (langlepsi+chi|psi+chi
angle)=(langlepsi|psi
angle)+(langlepsi|chi
angle)+(langlechi|psi
angle)+(langlechi|chi
angle).
Exercise 2.1 Consider the two states|=i|1+3i|2-|3
angle and|x=|1-i|2+5i|3
angle whereandare orthonormal. aCalculate(w|),(x|x),|x,x
angle, and infer(w+x|w+x).Are the scalar products( | x) and(x |w) equal?
Hint: Find the corresponding bras < and< first. Then take the scalar(inner product remembering that <|>=<|>=<|>=1 and all other scalar (inner) products are zero.
Also,<+x+x>=<>+<|x>+<>+<x>